Implicit plots#

sage.plot.plot3d.implicit_plot3d.implicit_plot3d(f, xrange, yrange, zrange, **kwds)[source]#

Plot an isosurface of a function.

INPUT:

  • f – function

  • xrange – a 2-tuple (x_min, x_max) or a 3-tuple (x, x_min, x_max)

  • yrange – a 2-tuple (y_min, y_max) or a 3-tuple (y, y_min, y_max)

  • zrange – a 2-tuple (z_min, z_max) or a 3-tuple (z, z_min, z_max)

  • plot_points – (default: “automatic”, which is 40) the number of function evaluations in each direction. (The number of cubes in the marching cubes algorithm will be one less than this). Can be a triple of integers, to specify a different resolution in each of x,y,z.

  • contour – (default: 0) plot the isosurface f(x,y,z) == contour. Can be a list, in which case multiple contours are plotted.

  • region – (default: None) If region is given, it must be a Python callable. Only segments of the surface where region(x,y,z) returns a number \(>0\) will be included in the plot. (Note that returning a Python boolean is acceptable, since True == 1 and False == 0).

EXAMPLES:

sage: var('x,y,z')
(x, y, z)

>>> from sage.all import *
>>> var('x,y,z')
(x, y, z)

A simple sphere:

sage: implicit_plot3d(x^2+y^2+z^2==4, (x,-3,3), (y,-3,3), (z,-3,3))
Graphics3d Object

>>> from sage.all import *
>>> implicit_plot3d(x**Integer(2)+y**Integer(2)+z**Integer(2)==Integer(4), (x,-Integer(3),Integer(3)), (y,-Integer(3),Integer(3)), (z,-Integer(3),Integer(3)))
Graphics3d Object
../../../_images/implicit_plot3d-1.svg

A nested set of spheres with a hole cut out:

sage: F = x^2 + y^2 + z^2
sage: P = implicit_plot3d(F, (x,-2,2), (y,-2,2), (z,-2,2), plot_points=60,
....:                     contour=[1,3,5], color='aquamarine',
....:                     region=lambda x,y,z: x<=0.2 or y>=0.2 or z<=0.2)
sage: P.show(viewer='tachyon')  # long time

>>> from sage.all import *
>>> F = x**Integer(2) + y**Integer(2) + z**Integer(2)
>>> P = implicit_plot3d(F, (x,-Integer(2),Integer(2)), (y,-Integer(2),Integer(2)), (z,-Integer(2),Integer(2)), plot_points=Integer(60),
...                     contour=[Integer(1),Integer(3),Integer(5)], color='aquamarine',
...                     region=lambda x,y,z: x<=RealNumber('0.2') or y>=RealNumber('0.2') or z<=RealNumber('0.2'))
>>> P.show(viewer='tachyon')  # long time
../../../_images/implicit_plot3d-2.svg

A very pretty example, attributed to Douglas Summers-Stay (archived page):

sage: T = RDF(golden_ratio)
sage: F = 2 - (cos(x+T*y) + cos(x-T*y) + cos(y+T*z)
....:           + cos(y-T*z) + cos(z-T*x) + cos(z+T*x))
sage: r = 4.77
sage: implicit_plot3d(F, (x,-r,r), (y,-r,r), (z,-r,r),
....:                 plot_points=40, color='darkkhaki').show(viewer='tachyon')

>>> from sage.all import *
>>> T = RDF(golden_ratio)
>>> F = Integer(2) - (cos(x+T*y) + cos(x-T*y) + cos(y+T*z)
...           + cos(y-T*z) + cos(z-T*x) + cos(z+T*x))
>>> r = RealNumber('4.77')
>>> implicit_plot3d(F, (x,-r,r), (y,-r,r), (z,-r,r),
...                 plot_points=Integer(40), color='darkkhaki').show(viewer='tachyon')
../../../_images/implicit_plot3d-3.svg

As I write this (but probably not as you read it), it’s almost Valentine’s day, so let’s try a heart (from http://mathworld.wolfram.com/HeartSurface.html)

sage: F = (x^2+9/4*y^2+z^2-1)^3 - x^2*z^3 - 9/(80)*y^2*z^3
sage: r = 1.5
sage: V = implicit_plot3d(F, (x,-r,r), (y,-r,r), (z,-r,r),
....:                     plot_points=80, color='red', smooth=False)
sage: V.show(viewer='tachyon')  # long time

>>> from sage.all import *
>>> F = (x**Integer(2)+Integer(9)/Integer(4)*y**Integer(2)+z**Integer(2)-Integer(1))**Integer(3) - x**Integer(2)*z**Integer(3) - Integer(9)/(Integer(80))*y**Integer(2)*z**Integer(3)
>>> r = RealNumber('1.5')
>>> V = implicit_plot3d(F, (x,-r,r), (y,-r,r), (z,-r,r),
...                     plot_points=Integer(80), color='red', smooth=False)
>>> V.show(viewer='tachyon')  # long time
../../../_images/implicit_plot3d-4.svg

The same examples also work with the default Jmol viewer; for example:

sage: T = RDF(golden_ratio)
sage: F = 2 - (cos(x + T*y) + cos(x - T*y) + cos(y + T*z)
....:           + cos(y - T*z) + cos(z - T*x) + cos(z + T*x))
sage: r = 4.77
sage: implicit_plot3d(F, (x,-r,r), (y,-r,r), (z,-r,r),
....:                 plot_points=40, color='deepskyblue').show()

>>> from sage.all import *
>>> T = RDF(golden_ratio)
>>> F = Integer(2) - (cos(x + T*y) + cos(x - T*y) + cos(y + T*z)
...           + cos(y - T*z) + cos(z - T*x) + cos(z + T*x))
>>> r = RealNumber('4.77')
>>> implicit_plot3d(F, (x,-r,r), (y,-r,r), (z,-r,r),
...                 plot_points=Integer(40), color='deepskyblue').show()

Here we use smooth=True with a Tachyon graph:

sage: implicit_plot3d(x^2 + y^2 + z^2, (x,-2,2), (y,-2,2), (z,-2,2),
....:                 contour=4, color='deepskyblue', smooth=True)
Graphics3d Object

>>> from sage.all import *
>>> implicit_plot3d(x**Integer(2) + y**Integer(2) + z**Integer(2), (x,-Integer(2),Integer(2)), (y,-Integer(2),Integer(2)), (z,-Integer(2),Integer(2)),
...                 contour=Integer(4), color='deepskyblue', smooth=True)
Graphics3d Object
../../../_images/implicit_plot3d-5.svg

We explicitly specify a gradient function (in conjunction with smooth=True) and invert the normals:

sage: gx = lambda x, y, z: -(2*x + y^2 + z^2)
sage: gy = lambda x, y, z: -(x^2 + 2*y + z^2)
sage: gz = lambda x, y, z: -(x^2 + y^2 + 2*z)
sage: implicit_plot3d(x^2+y^2+z^2, (x,-2,2), (y,-2,2), (z,-2,2),
....:                 contour=4, plot_points=40, smooth=True,
....:                 gradient=(gx, gy, gz)).show(viewer='tachyon')

>>> from sage.all import *
>>> gx = lambda x, y, z: -(Integer(2)*x + y**Integer(2) + z**Integer(2))
>>> gy = lambda x, y, z: -(x**Integer(2) + Integer(2)*y + z**Integer(2))
>>> gz = lambda x, y, z: -(x**Integer(2) + y**Integer(2) + Integer(2)*z)
>>> implicit_plot3d(x**Integer(2)+y**Integer(2)+z**Integer(2), (x,-Integer(2),Integer(2)), (y,-Integer(2),Integer(2)), (z,-Integer(2),Integer(2)),
...                 contour=Integer(4), plot_points=Integer(40), smooth=True,
...                 gradient=(gx, gy, gz)).show(viewer='tachyon')
../../../_images/implicit_plot3d-6.svg

A graph of two metaballs interacting with each other:

sage: def metaball(x0, y0, z0):
....:     return 1 / ((x-x0)^2+(y-y0)^2+(z-z0)^2)
sage: implicit_plot3d(metaball(-0.6,0,0) + metaball(0.6,0,0),
....:                 (x,-2,2), (y,-2,2), (z,-2,2),
....:                 plot_points=60, contour=2, color='seagreen')
Graphics3d Object

>>> from sage.all import *
>>> def metaball(x0, y0, z0):
...     return Integer(1) / ((x-x0)**Integer(2)+(y-y0)**Integer(2)+(z-z0)**Integer(2))
>>> implicit_plot3d(metaball(-RealNumber('0.6'),Integer(0),Integer(0)) + metaball(RealNumber('0.6'),Integer(0),Integer(0)),
...                 (x,-Integer(2),Integer(2)), (y,-Integer(2),Integer(2)), (z,-Integer(2),Integer(2)),
...                 plot_points=Integer(60), contour=Integer(2), color='seagreen')
Graphics3d Object
../../../_images/implicit_plot3d-7.svg

One can also color the surface using a coloring function and a colormap as follows. Note that the coloring function must take values in the interval \([0,1]\).

sage: t = (sin(3*z)**2).function(x,y,z)
sage: cm = colormaps.gist_rainbow
sage: G = implicit_plot3d(x^2 + y^2 + z^2, (x,-2,2), (y,-2,2), (z,-2, 2),
....:                     contour=4, color=(t,cm), plot_points=100)
sage: G.show(viewer='tachyon')  # long time

>>> from sage.all import *
>>> t = (sin(Integer(3)*z)**Integer(2)).function(x,y,z)
>>> cm = colormaps.gist_rainbow
>>> G = implicit_plot3d(x**Integer(2) + y**Integer(2) + z**Integer(2), (x,-Integer(2),Integer(2)), (y,-Integer(2),Integer(2)), (z,-Integer(2), Integer(2)),
...                     contour=Integer(4), color=(t,cm), plot_points=Integer(100))
>>> G.show(viewer='tachyon')  # long time
../../../_images/implicit_plot3d-8.svg

Here is another colored example:

sage: x, y, z = var('x,y,z')
sage: t = (x).function(x,y,z)
sage: cm = colormaps.PiYG
sage: G = implicit_plot3d(x^4 + y^2 + z^2, (x,-2,2), (y,-2,2),(z,-2,2),
....:                     contour=4, color=(t,cm), plot_points=40)
sage: G
Graphics3d Object

>>> from sage.all import *
>>> x, y, z = var('x,y,z')
>>> t = (x).function(x,y,z)
>>> cm = colormaps.PiYG
>>> G = implicit_plot3d(x**Integer(4) + y**Integer(2) + z**Integer(2), (x,-Integer(2),Integer(2)), (y,-Integer(2),Integer(2)),(z,-Integer(2),Integer(2)),
...                     contour=Integer(4), color=(t,cm), plot_points=Integer(40))
>>> G
Graphics3d Object
../../../_images/implicit_plot3d-9.svg

MANY MORE EXAMPLES:

A kind of saddle:

sage: implicit_plot3d(x^3 + y^2 - z^2, (x,-2,2), (y,-2,2), (z,-2,2),
....:                 plot_points=60, contour=0, color='lightcoral')
Graphics3d Object

>>> from sage.all import *
>>> implicit_plot3d(x**Integer(3) + y**Integer(2) - z**Integer(2), (x,-Integer(2),Integer(2)), (y,-Integer(2),Integer(2)), (z,-Integer(2),Integer(2)),
...                 plot_points=Integer(60), contour=Integer(0), color='lightcoral')
Graphics3d Object
../../../_images/implicit_plot3d-10.svg

A smooth surface with six radial openings:

sage: implicit_plot3d(-(cos(x) + cos(y) + cos(z)),
....:                 (x,-4,4), (y,-4,4), (z,-4,4), color='orchid')
Graphics3d Object

>>> from sage.all import *
>>> implicit_plot3d(-(cos(x) + cos(y) + cos(z)),
...                 (x,-Integer(4),Integer(4)), (y,-Integer(4),Integer(4)), (z,-Integer(4),Integer(4)), color='orchid')
Graphics3d Object
../../../_images/implicit_plot3d-11.svg

A cube composed of eight conjoined blobs:

sage: F = x^2 + y^2 + z^2 + cos(4*x) + cos(4*y) + cos(4*z) - 0.2
sage: implicit_plot3d(F, (x,-2,2), (y,-2,2), (z,-2,2), color='mediumspringgreen')
Graphics3d Object

>>> from sage.all import *
>>> F = x**Integer(2) + y**Integer(2) + z**Integer(2) + cos(Integer(4)*x) + cos(Integer(4)*y) + cos(Integer(4)*z) - RealNumber('0.2')
>>> implicit_plot3d(F, (x,-Integer(2),Integer(2)), (y,-Integer(2),Integer(2)), (z,-Integer(2),Integer(2)), color='mediumspringgreen')
Graphics3d Object
../../../_images/implicit_plot3d-12.svg

A variation of the blob cube featuring heterogeneously sized blobs:

sage: F = x^2 + y^2 + z^2 + sin(4*x) + sin(4*y) + sin(4*z) - 1
sage: implicit_plot3d(F, (x,-2,2), (y,-2,2), (z,-2,2), color='lavenderblush')
Graphics3d Object

>>> from sage.all import *
>>> F = x**Integer(2) + y**Integer(2) + z**Integer(2) + sin(Integer(4)*x) + sin(Integer(4)*y) + sin(Integer(4)*z) - Integer(1)
>>> implicit_plot3d(F, (x,-Integer(2),Integer(2)), (y,-Integer(2),Integer(2)), (z,-Integer(2),Integer(2)), color='lavenderblush')
Graphics3d Object
../../../_images/implicit_plot3d-13.svg

A Klein bottle:

sage: G = x^2 + y^2 + z^2
sage: F = (G+2*y-1)*((G-2*y-1)^2-8*z^2) + 16*x*z*(G-2*y-1)
sage: implicit_plot3d(F, (x,-3,3), (y,-3.1,3.1), (z,-4,4), color='moccasin')
Graphics3d Object

>>> from sage.all import *
>>> G = x**Integer(2) + y**Integer(2) + z**Integer(2)
>>> F = (G+Integer(2)*y-Integer(1))*((G-Integer(2)*y-Integer(1))**Integer(2)-Integer(8)*z**Integer(2)) + Integer(16)*x*z*(G-Integer(2)*y-Integer(1))
>>> implicit_plot3d(F, (x,-Integer(3),Integer(3)), (y,-RealNumber('3.1'),RealNumber('3.1')), (z,-Integer(4),Integer(4)), color='moccasin')
Graphics3d Object
../../../_images/implicit_plot3d-14.svg

A lemniscate:

sage: F = 4*x^2*(x^2+y^2+z^2+z) + y^2*(y^2+z^2-1)
sage: implicit_plot3d(F, (x,-0.5,0.5), (y,-1,1), (z,-1,1), color='deeppink')
Graphics3d Object

>>> from sage.all import *
>>> F = Integer(4)*x**Integer(2)*(x**Integer(2)+y**Integer(2)+z**Integer(2)+z) + y**Integer(2)*(y**Integer(2)+z**Integer(2)-Integer(1))
>>> implicit_plot3d(F, (x,-RealNumber('0.5'),RealNumber('0.5')), (y,-Integer(1),Integer(1)), (z,-Integer(1),Integer(1)), color='deeppink')
Graphics3d Object
../../../_images/implicit_plot3d-15.svg

Drope:

sage: implicit_plot3d(z - 4*x*exp(-x^2-y^2), (x,-2,2), (y,-2,2), (z,-1.7,1.7),
....:                 color='darkcyan')
Graphics3d Object

>>> from sage.all import *
>>> implicit_plot3d(z - Integer(4)*x*exp(-x**Integer(2)-y**Integer(2)), (x,-Integer(2),Integer(2)), (y,-Integer(2),Integer(2)), (z,-RealNumber('1.7'),RealNumber('1.7')),
...                 color='darkcyan')
Graphics3d Object
../../../_images/implicit_plot3d-16.svg

A cube with a circular aperture on each face:

sage: F = ((1/2.3)^2 * (x^2 + y^2 + z^2))^(-6) + ((1/2)^8 * (x^8 + y^8 + z^8))^6 - 1
sage: implicit_plot3d(F, (x,-2,2), (y,-2,2), (z,-2,2), color='palevioletred')
Graphics3d Object

>>> from sage.all import *
>>> F = ((Integer(1)/RealNumber('2.3'))**Integer(2) * (x**Integer(2) + y**Integer(2) + z**Integer(2)))**(-Integer(6)) + ((Integer(1)/Integer(2))**Integer(8) * (x**Integer(8) + y**Integer(8) + z**Integer(8)))**Integer(6) - Integer(1)
>>> implicit_plot3d(F, (x,-Integer(2),Integer(2)), (y,-Integer(2),Integer(2)), (z,-Integer(2),Integer(2)), color='palevioletred')
Graphics3d Object
../../../_images/implicit_plot3d-17.svg

A simple hyperbolic surface:

sage: implicit_plot3d(x^2 + y - z^2, (x,-1,1), (y,-1,1), (z,-1,1),
....:                 color='darkslategray')
Graphics3d Object

>>> from sage.all import *
>>> implicit_plot3d(x**Integer(2) + y - z**Integer(2), (x,-Integer(1),Integer(1)), (y,-Integer(1),Integer(1)), (z,-Integer(1),Integer(1)),
...                 color='darkslategray')
Graphics3d Object
../../../_images/implicit_plot3d-18.svg

A hyperboloid:

sage: implicit_plot3d(x^2 + y^2 - z^2 -0.3, (x,-2,2), (y,-2,2), (z,-1.8,1.8),
....:                 color='honeydew')
Graphics3d Object

>>> from sage.all import *
>>> implicit_plot3d(x**Integer(2) + y**Integer(2) - z**Integer(2) -RealNumber('0.3'), (x,-Integer(2),Integer(2)), (y,-Integer(2),Integer(2)), (z,-RealNumber('1.8'),RealNumber('1.8')),
...                 color='honeydew')
Graphics3d Object
../../../_images/implicit_plot3d-19.svg

Dupin cyclide (Wikipedia article Dupin_cyclide)

sage: x, y, z, a, b, c, d = var('x,y,z,a,b,c,d')
sage: a = 3.5
sage: b = 3
sage: c = sqrt(a^2 - b^2)
sage: d = 2
sage: F = (x^2 + y^2 + z^2 + b^2 - d^2)^2 - 4*(a*x-c*d)^2 - 4*b^2*y^2
sage: implicit_plot3d(F, (x,-6,6), (y,-6,6), (z,-6,6), color='seashell')
Graphics3d Object

>>> from sage.all import *
>>> x, y, z, a, b, c, d = var('x,y,z,a,b,c,d')
>>> a = RealNumber('3.5')
>>> b = Integer(3)
>>> c = sqrt(a**Integer(2) - b**Integer(2))
>>> d = Integer(2)
>>> F = (x**Integer(2) + y**Integer(2) + z**Integer(2) + b**Integer(2) - d**Integer(2))**Integer(2) - Integer(4)*(a*x-c*d)**Integer(2) - Integer(4)*b**Integer(2)*y**Integer(2)
>>> implicit_plot3d(F, (x,-Integer(6),Integer(6)), (y,-Integer(6),Integer(6)), (z,-Integer(6),Integer(6)), color='seashell')
Graphics3d Object
../../../_images/implicit_plot3d-20.svg

Sinus:

sage: implicit_plot3d(sin(pi*((x)^2+(y)^2))/2 + z, (x,-1,1), (y,-1,1), (z,-1,1),
....:                 color='rosybrown')
Graphics3d Object

>>> from sage.all import *
>>> implicit_plot3d(sin(pi*((x)**Integer(2)+(y)**Integer(2)))/Integer(2) + z, (x,-Integer(1),Integer(1)), (y,-Integer(1),Integer(1)), (z,-Integer(1),Integer(1)),
...                 color='rosybrown')
Graphics3d Object
../../../_images/implicit_plot3d-21.svg

A torus:

sage: implicit_plot3d((sqrt(x*x+y*y)-3)^2 + z*z - 1, (x,-4,4), (y,-4,4), (z,-1,1),
....:                 color='indigo')
Graphics3d Object

>>> from sage.all import *
>>> implicit_plot3d((sqrt(x*x+y*y)-Integer(3))**Integer(2) + z*z - Integer(1), (x,-Integer(4),Integer(4)), (y,-Integer(4),Integer(4)), (z,-Integer(1),Integer(1)),
...                 color='indigo')
Graphics3d Object
../../../_images/implicit_plot3d-22.svg

An octahedron:

sage: implicit_plot3d(abs(x) + abs(y) + abs(z) - 1, (x,-1,1), (y,-1,1), (z,-1,1),
....:                 color='olive')
Graphics3d Object

>>> from sage.all import *
>>> implicit_plot3d(abs(x) + abs(y) + abs(z) - Integer(1), (x,-Integer(1),Integer(1)), (y,-Integer(1),Integer(1)), (z,-Integer(1),Integer(1)),
...                 color='olive')
Graphics3d Object
../../../_images/implicit_plot3d-23.svg

A cube:

sage: implicit_plot3d(x^100 + y^100 + z^100 - 1, (x,-2,2), (y,-2,2), (z,-2,2),
....:                 color='lightseagreen')
Graphics3d Object

>>> from sage.all import *
>>> implicit_plot3d(x**Integer(100) + y**Integer(100) + z**Integer(100) - Integer(1), (x,-Integer(2),Integer(2)), (y,-Integer(2),Integer(2)), (z,-Integer(2),Integer(2)),
...                 color='lightseagreen')
Graphics3d Object
../../../_images/implicit_plot3d-24.svg

Toupie:

sage: implicit_plot3d((sqrt(x*x+y*y)-3)^3 + z*z - 1, (x,-4,4), (y,-4,4), (z,-6,6),
....:                 color='mintcream')
Graphics3d Object

>>> from sage.all import *
>>> implicit_plot3d((sqrt(x*x+y*y)-Integer(3))**Integer(3) + z*z - Integer(1), (x,-Integer(4),Integer(4)), (y,-Integer(4),Integer(4)), (z,-Integer(6),Integer(6)),
...                 color='mintcream')
Graphics3d Object
../../../_images/implicit_plot3d-25.svg

A cube with rounded edges:

sage: F = x^4 + y^4 + z^4 - (x^2 + y^2 + z^2)
sage: implicit_plot3d(F, (x,-2,2), (y,-2,2), (z,-2,2), color='mediumvioletred')
Graphics3d Object

>>> from sage.all import *
>>> F = x**Integer(4) + y**Integer(4) + z**Integer(4) - (x**Integer(2) + y**Integer(2) + z**Integer(2))
>>> implicit_plot3d(F, (x,-Integer(2),Integer(2)), (y,-Integer(2),Integer(2)), (z,-Integer(2),Integer(2)), color='mediumvioletred')
Graphics3d Object
../../../_images/implicit_plot3d-26.svg

Chmutov:

sage: F = x^4 + y^4 + z^4 - (x^2 + y^2 + z^2 - 0.3)
sage: implicit_plot3d(F, (x,-1.5,1.5), (y,-1.5,1.5), (z,-1.5,1.5),
....:                 color='lightskyblue')
Graphics3d Object

>>> from sage.all import *
>>> F = x**Integer(4) + y**Integer(4) + z**Integer(4) - (x**Integer(2) + y**Integer(2) + z**Integer(2) - RealNumber('0.3'))
>>> implicit_plot3d(F, (x,-RealNumber('1.5'),RealNumber('1.5')), (y,-RealNumber('1.5'),RealNumber('1.5')), (z,-RealNumber('1.5'),RealNumber('1.5')),
...                 color='lightskyblue')
Graphics3d Object
../../../_images/implicit_plot3d-27.svg

Further Chmutov:

sage: F = 2*(x^2*(3-4*x^2)^2+y^2*(3-4*y^2)^2+z^2*(3-4*z^2)^2) - 3
sage: implicit_plot3d(F, (x,-1.3,1.3), (y,-1.3,1.3), (z,-1.3,1.3),
....:                 color='darksalmon')
Graphics3d Object

>>> from sage.all import *
>>> F = Integer(2)*(x**Integer(2)*(Integer(3)-Integer(4)*x**Integer(2))**Integer(2)+y**Integer(2)*(Integer(3)-Integer(4)*y**Integer(2))**Integer(2)+z**Integer(2)*(Integer(3)-Integer(4)*z**Integer(2))**Integer(2)) - Integer(3)
>>> implicit_plot3d(F, (x,-RealNumber('1.3'),RealNumber('1.3')), (y,-RealNumber('1.3'),RealNumber('1.3')), (z,-RealNumber('1.3'),RealNumber('1.3')),
...                 color='darksalmon')
Graphics3d Object
../../../_images/implicit_plot3d-28.svg

Clebsch surface:

sage: F_1 = 81 * (x^3+y^3+z^3)
sage: F_2 = 189 * (x^2*(y+z)+y^2*(x+z)+z^2*(x+y))
sage: F_3 = 54 * x * y * z
sage: F_4 = 126 * (x*y+x*z+y*z)
sage: F_5 = 9 * (x^2+y^2+z^2)
sage: F_6 = 9 * (x+y+z)
sage: F = F_1 - F_2 + F_3 + F_4 - F_5 + F_6 + 1
sage: implicit_plot3d(F, (x,-1,1), (y,-1,1), (z,-1,1), color='yellowgreen')
Graphics3d Object

>>> from sage.all import *
>>> F_1 = Integer(81) * (x**Integer(3)+y**Integer(3)+z**Integer(3))
>>> F_2 = Integer(189) * (x**Integer(2)*(y+z)+y**Integer(2)*(x+z)+z**Integer(2)*(x+y))
>>> F_3 = Integer(54) * x * y * z
>>> F_4 = Integer(126) * (x*y+x*z+y*z)
>>> F_5 = Integer(9) * (x**Integer(2)+y**Integer(2)+z**Integer(2))
>>> F_6 = Integer(9) * (x+y+z)
>>> F = F_1 - F_2 + F_3 + F_4 - F_5 + F_6 + Integer(1)
>>> implicit_plot3d(F, (x,-Integer(1),Integer(1)), (y,-Integer(1),Integer(1)), (z,-Integer(1),Integer(1)), color='yellowgreen')
Graphics3d Object
../../../_images/implicit_plot3d-29.svg

Looks like a water droplet:

sage: implicit_plot3d(x^2 + y^2 - (1-z)*z^2, (x,-1.5,1.5), (y,-1.5,1.5), (z,-1,1),
....:                 color='bisque')
Graphics3d Object

>>> from sage.all import *
>>> implicit_plot3d(x**Integer(2) + y**Integer(2) - (Integer(1)-z)*z**Integer(2), (x,-RealNumber('1.5'),RealNumber('1.5')), (y,-RealNumber('1.5'),RealNumber('1.5')), (z,-Integer(1),Integer(1)),
...                 color='bisque')
Graphics3d Object
../../../_images/implicit_plot3d-30.svg

Sphere in a cage:

sage: F = (x^8+z^30+y^8-(x^4 + z^50 + y^4 -0.3)) * (x^2+y^2+z^2-0.5)
sage: implicit_plot3d(F, (x,-1.2,1.2), (y,-1.3,1.3), (z,-1.5,1.5),
....:                 color='firebrick')
Graphics3d Object

>>> from sage.all import *
>>> F = (x**Integer(8)+z**Integer(30)+y**Integer(8)-(x**Integer(4) + z**Integer(50) + y**Integer(4) -RealNumber('0.3'))) * (x**Integer(2)+y**Integer(2)+z**Integer(2)-RealNumber('0.5'))
>>> implicit_plot3d(F, (x,-RealNumber('1.2'),RealNumber('1.2')), (y,-RealNumber('1.3'),RealNumber('1.3')), (z,-RealNumber('1.5'),RealNumber('1.5')),
...                 color='firebrick')
Graphics3d Object
../../../_images/implicit_plot3d-31.svg

Ortho circle:

sage: F = (((x^2+y^2-1)^2+z^2) * ((y^2+z^2-1)^2+x^2) * ((z^2+x^2-1)^2+y^2)
....:      - 0.075^2 * (1+3*(x^2+y^2+z^2)))
sage: implicit_plot3d(F, (x,-1.5,1.5), (y,-1.5,1.5), (z,-1.5,1.5),
....:                 color='lemonchiffon')
Graphics3d Object

>>> from sage.all import *
>>> F = (((x**Integer(2)+y**Integer(2)-Integer(1))**Integer(2)+z**Integer(2)) * ((y**Integer(2)+z**Integer(2)-Integer(1))**Integer(2)+x**Integer(2)) * ((z**Integer(2)+x**Integer(2)-Integer(1))**Integer(2)+y**Integer(2))
...      - RealNumber('0.075')**Integer(2) * (Integer(1)+Integer(3)*(x**Integer(2)+y**Integer(2)+z**Integer(2))))
>>> implicit_plot3d(F, (x,-RealNumber('1.5'),RealNumber('1.5')), (y,-RealNumber('1.5'),RealNumber('1.5')), (z,-RealNumber('1.5'),RealNumber('1.5')),
...                 color='lemonchiffon')
Graphics3d Object
../../../_images/implicit_plot3d-32.svg

Cube sphere:

sage: F = 12 - ((1/2.3)^2 *(x^2 + y^2 + z^2))^-6 - ((1/2)^8 * (x^8 + y^8 + z^8))^6
sage: implicit_plot3d(F, (x,-2,2), (y,-2,2), (z,-2,2), color='rosybrown')
Graphics3d Object

>>> from sage.all import *
>>> F = Integer(12) - ((Integer(1)/RealNumber('2.3'))**Integer(2) *(x**Integer(2) + y**Integer(2) + z**Integer(2)))**-Integer(6) - ((Integer(1)/Integer(2))**Integer(8) * (x**Integer(8) + y**Integer(8) + z**Integer(8)))**Integer(6)
>>> implicit_plot3d(F, (x,-Integer(2),Integer(2)), (y,-Integer(2),Integer(2)), (z,-Integer(2),Integer(2)), color='rosybrown')
Graphics3d Object
../../../_images/implicit_plot3d-33.svg

Two cylinders intersect to make a cross:

sage: implicit_plot3d((x^2+y^2-1) * (x^2+z^2-1) - 1, (x,-3,3), (y,-3,3), (z,-3,3),
....:                 color='burlywood')
Graphics3d Object

>>> from sage.all import *
>>> implicit_plot3d((x**Integer(2)+y**Integer(2)-Integer(1)) * (x**Integer(2)+z**Integer(2)-Integer(1)) - Integer(1), (x,-Integer(3),Integer(3)), (y,-Integer(3),Integer(3)), (z,-Integer(3),Integer(3)),
...                 color='burlywood')
Graphics3d Object
../../../_images/implicit_plot3d-34.svg

Three cylinders intersect in a similar fashion:

sage: implicit_plot3d((x^2+y^2-1) * (x^2+z^2-1) * (y^2+z^2-1) - 1,
....:                 (x,-3,3), (y,-3,3), (z,-3,3), color='aqua')
Graphics3d Object

>>> from sage.all import *
>>> implicit_plot3d((x**Integer(2)+y**Integer(2)-Integer(1)) * (x**Integer(2)+z**Integer(2)-Integer(1)) * (y**Integer(2)+z**Integer(2)-Integer(1)) - Integer(1),
...                 (x,-Integer(3),Integer(3)), (y,-Integer(3),Integer(3)), (z,-Integer(3),Integer(3)), color='aqua')
Graphics3d Object
../../../_images/implicit_plot3d-35.svg

A sphere-ish object with twelve holes, four on each XYZ plane:

sage: implicit_plot3d(3*(cos(x)+cos(y)+cos(z)) + 4*cos(x)*cos(y)*cos(z),
....:                 (x,-3,3), (y,-3,3), (z,-3,3), color='orangered')
Graphics3d Object

>>> from sage.all import *
>>> implicit_plot3d(Integer(3)*(cos(x)+cos(y)+cos(z)) + Integer(4)*cos(x)*cos(y)*cos(z),
...                 (x,-Integer(3),Integer(3)), (y,-Integer(3),Integer(3)), (z,-Integer(3),Integer(3)), color='orangered')
Graphics3d Object
../../../_images/implicit_plot3d-36.svg

A gyroid:

sage: implicit_plot3d(cos(x)*sin(y) + cos(y)*sin(z) + cos(z)*sin(x),
....:                 (x,-4,4), (y,-4,4), (z,-4,4), color='sandybrown')
Graphics3d Object

>>> from sage.all import *
>>> implicit_plot3d(cos(x)*sin(y) + cos(y)*sin(z) + cos(z)*sin(x),
...                 (x,-Integer(4),Integer(4)), (y,-Integer(4),Integer(4)), (z,-Integer(4),Integer(4)), color='sandybrown')
Graphics3d Object
../../../_images/implicit_plot3d-37.svg

Tetrahedra:

sage: implicit_plot3d((x^2+y^2+z^2)^2 + 8*x*y*z - 10*(x^2+y^2+z^2) + 25,
....:                 (x,-4,4), (y,-4,4), (z,-4,4), color='plum')
Graphics3d Object

>>> from sage.all import *
>>> implicit_plot3d((x**Integer(2)+y**Integer(2)+z**Integer(2))**Integer(2) + Integer(8)*x*y*z - Integer(10)*(x**Integer(2)+y**Integer(2)+z**Integer(2)) + Integer(25),
...                 (x,-Integer(4),Integer(4)), (y,-Integer(4),Integer(4)), (z,-Integer(4),Integer(4)), color='plum')
Graphics3d Object
../../../_images/implicit_plot3d-38.svg